Electrical responses in physical structures to electrical forces (e.g., capacitive response to varying voltage or inductive response to varying current) are well known and circuits are easily modeled. Electromagnetic responses in these same physical structures are equally well known, but are much less easily modeled. Thus, electrical circuit analogs are typically used to model electromagnetic responses. One well-known circuit based approach for modeling electromagnetic responses is known as the Partial Element Equivalent Circuit (PEEC) model. A typical PEEC model is numerically equivalent to a full wave method like the method of moments (MoM) solution with Galerkin matching. Simple PEEC models (that do not involve delays) provide an adequate frequency response approximation in a well-defined low frequency range. PEEC models may be used, for example, to model interconnects in Integrated Circuit (IC) package wiring and inter-package connections. Further, these simple PEEC models have been used to model simple and complex interconnect conditions in a conventional circuit modeling program, e.g., SPICE or ASTAP, for both time and frequency domain modeling.
However, typical state of the art PEEC models (as well as other time domain integral equation models/solutions) are known to include instabilities above factive, also known as late time instability where factive is the highest frequency of interest for the modeling. Resistors, inductors and capacitors model as ideal elements (i.e., pure resistance, pure inductance and pure capacitance), where ideal elements do not exist in nature. Further, an LC network or an RLC network only provides an analogous electrical-response approximation to electromagnetic effects. Thus, the instabilities are clearly non-physical, i.e., although the instabilities do not occur in the physical structure being modeled, they are inherent in the circuit model.
Unfortunately, previous attempts to eliminate this time domain instability have met with only partial success. Models utilizing implicit time integration methods such as the backward Euler formula, for example, have proven more stable than explicit method based models like the forward Euler method. Another partially successful example is coefficient subdivision (+PEEC), where the useful frequency range or active range, factive, is extended by subdividing or segmenting the partial elements in the PEEC model into smaller and smaller segments. Also, dampening resistors may be included (R+PEEC), e.g., in parallel with calculated partial inductances, to reduce model resonance above factive. Consequently, these approaches have only marginally increased PEEC model stability.
This high frequency instability problem causes an exponential increase in the time domain response. This exponential response increase is accompanied by an increase in computer processing time, i.e., calculating the exponentially increasing response is accompanied by an increase in computer resources needed to converge on a solution. Prior art approaches to addressing this instability have also increased demand for computer resources. For example, coefficient subdivision requires further segmenting the model to extend the frequency range with each additional segment proportionately increasing computer resources needed. Including damping resistors further adds to the network complexity and, accordingly, increases computer resources needed for processing the increasingly complex model. Consequently, problems with this anomalous high frequency response have offset many of the advantages of using this otherwise very efficient modeling approach (+PEEC or R+PEEC) for these normally (even well below factive) complex time domain problems, e.g., modeling electromagnetic responses in interconnects. Further, for higher performance applications, where the switching signal spectrum approaches factive and, correspondingly, signal edge rise and fall times decrease dramatically, model instabilities are inevitable using these prior art electromagnetic models.
Thus, there is a need for a model for electromagnetic responses in physical structures that is free of instabilities in the extended frequency range, even beyond the typical useful frequency range or active range (factive), and especially for such a PEEC model.